Exchange graphs

Visual reference: B. H. An, Y. Bae and E. Lee, Lagrangian fillings for Legendrian links of finite or affine Dynkin type, §2.4, Example 2.34 and Figure 12. The D₄ view uses the cube F₋α₂ as the outer facet and colors the other initial facets. These are newly computed drawings; their coordinates need not match the paper.

Construction: F. Chapoton, S. Fomin and A. Zelevinsky, Polytopal realizations of generalized associahedra, Theorem 1.5 and Corollary 1.9. Vertices are maximal compatible sets of almost positive roots. Edges exchange exactly one root. The polytope is defined by ⟨β,z⟩ ≤ F(β), with F invariant under τ₊, τ₋ and F(−αᵢ) taken from ρ∨. Rank 4 is shown by a Schlegel projection into a selected 3-dimensional facet, followed by an ordinary 3D view. Crossings on screen are not extra vertices.

Root convention: sᵢ(αⱼ) = αⱼ − aᵢⱼαᵢ. The initial cluster is {−α₁,…,−αₙ}; μQ acts as τ₊τ₋ with vertex 1 in I₊. Bₙ and Cₙ have isomorphic exchange graphs, but different root labels. Both use textbook numbering: α₁ is the simple root of different length, short in Bₙ and long in Cₙ. Only irreducible types are listed; products of types are not included.

Polygon model: Fomin–Williams–Zelevinsky, Chapters 1–3, Figure 1.4. Aₘ clusters are triangulations of an (m+3)-gon. The initial diagonals realize the same bipartite seed as the root atlas; boundary sides specialize to 1. The A₅ and A₇ extensions use a 3D linear projection of higher-dimensional vertex coordinates, not a Schlegel diagram. Folding follows §4.4 and §5.5: row-orbit sums give Cₙ in the textbook convention. The folded matrix places the singleton diameter orbit first: α₁ is long and α₂, …, αₙ are short in the initial Cₙ seed (n ≥ 2). Centrally symmetric triangulations are vertices of A₂ₙ₋₁, but a paired flip is a two-edge path through a nonsymmetric triangulation, not a single A-type edge. Dashed orange links represent these orbit mutations; they are not additional edges of the A-type associahedron.

Seeds use no frozen variables and all coefficients are 1. Initial exchange matrix: bᵢⱼ = −aᵢⱼ when i ∈ I₊ and bᵢⱼ = aᵢⱼ when i ∈ I₋ (i ≠ j). An arrow i → j means bᵢⱼ > 0. For a valued arrow, the label (p,q) gives (bᵢⱼ, −bⱼᵢ) in arrow direction. Xᵢ denotes a current seed slot; xᵢ denotes an initial variable. Slot order is preserved along mutations; a direct jump to a nonadjacent cluster uses its canonical root order. Returning around a loop can permute seed slots without changing the cluster.